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**laipou****Member**- Registered: 2009-09-19
- Posts: 24

Which roots of unity are contained in

,where is a square free integer?Thanks for any help.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

d > 0 is easy.

For d < 0, we have that d^2 is in Q. Thus, all polynomials can be represented by a + bd, where a and b are rationals. Now you want to find all elements of this form z where z^n = 1.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**laipou****Member**- Registered: 2009-09-19
- Posts: 24

Thank you for you suggestion,Ricky.

Here is the answer I think.

Let

Then we know (Euler function).

Since ,if,then we must have or.

Finaly,we get .

.

o,w.

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